introduction to discrete time signals & system

8/19/2019 Introduction to Discrete Time Signals & System

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apter 1

roduction to Discrete Time Signals & System:

Discrete–Time Signals representation and Manipulation,

Discrete–Time IIR and FIR Systems, Impulse Response,

(Infinite Impulse Response and Finite Impulse Response)

Transer Function,

Dierence !"uation,

Fre"uency Domain and Time Domain #nalysis o IIR ilter and FIR ilter,

Correlation,

$inear and Circular and Con%olution #lgorithm,


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pter 1 'uestions

ec( )1

i%e any i%e classiications o Discrete time systems +ith eamples

.t/ 0 sin.2π

t/ 3 4 sin .5)π

t/ is sampled +ith Fs 0 6 times per sec.

1/ 7hat are the re"uencies in radians in the resulting DT signal .n/8

)/ I .n/ is passed through an ideal interpolator,

+hat is the reconstructed signal8

une )11

.n/ 0 ), ;1, 4, , < o=tain ollo+ing:

.i/ .;n/ .ii/ .n;1/ .iii/ .n31/ .i%/ .;n3)/.%/ .)n/

or a discrete time system +hose impulse response h .n/ 0 1, ;), 1<

↑

ind the output or input .n/ 0 1, ), 4, <

↑

lassiy ollo+ing DT System on linearity> causality and time %ariance:;

i/ y .n/ 0 ).n/ 3 .n;1/ .ii/ y .n/ 0 .)n/ 3)


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1( .n/0 ?1, ), 1, ), 4, @ ind y .n/ 0 .1;n/ 3 .4;n/ 3 . –n/

)( !plain +hether the ollo+ing signals are po+er signal or energy signal(

a( (A n u .n/ =( # cos .ω

n/

4( Determine +hether the ollo+ing signals are periodic or non;periodic(

B.π

>/ n

a( .n/ 0 e =( .n/ 0 cos.A πt/ 3 cos.1 πt/

( Decompose the ollo+ing signals into e%en and odd parts

a( ?n@ 0 ? 1 ) 4 A 6 5 2 @

=( y ?n@ 0 ?2 2 2 2 @


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Signal

# signal is deined as any physical "uantity that %aries +ith:

1( Time,

)( Space or

4( #ny other independent %aria=le or %aria=les

; !amples o Signals

; Speech

; Music

; ictures; ideo

; !C*

; Mathematically, +e descri=e a signal as a

unction o one or more independent %aria=les

s1.t / 0 A t

s). t/ 0 ) t) Ene independent %aria=le t .time/

., y/ 0 4 3 ) y 3 1 y)

t+o independent %aria=les , and y


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!ample o speech signal

Signals can =e generated =y a system:

Speech ; ocal Cords

!C* ; !lectrocardiogram: olariation> depolariation o %entricles

!!* ; !lectroencephalogram


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; Speech, !C* and !!* signals are unction o a single %aria=le:

t, Time

; #n image signal is unction o t+o %aria=les:

, y .Coordinates o an image/

; These signals are generated =y some means:

; Speech signal =y %ocal cord and air lo+(

; Image signals =y eposing light; sensiti%e sensors to light

; The signals are generated =y a system


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Signal processing system

# signal processing system is deined as a de%ice

that perorm an operation on the signal

; #mpliier is a system that ampliies a signal

; Filter is a system that suppresses or allo+s

some re"uencies rom the signal

; Thus a system is an interconnection o components thatperorms an operation on

input signal and produces an output signal(

; The deinition o a system can =e =roadened to include not only

physical de%ices =ut also,

sot+are realiation o operations


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Digital signal processing system

; There are t+o types o signals:

; #nalog signals and

; Digital signals

#nalog signals:

; Most o the signals that occur in nature are analog signals(

; The analog signals are a unction o a continuous %aria=le

such as time or space,

taGes on %alues in continuous range

; #nalog signals can =e processed directly, in continuous orm =y analog systems .circuits/ such as:

#mpliiers, Filters, Fre"uency multipliers,

; #nalog processing:

Direct processing o signals in continuous orm .#nalog orm/


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!ample o analog processing o signals

oice ampliier:

; #mpliier is made up o resistors, capacitors, transistors etc(

; #mpliier taGes the analog signal .continuous/ rom microphone,

ampliies it , and produces analog output signal or speaGer

#nalog processing system:

; TaGes analog input, process it in analog orm and

produces analog output


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Digital Signal rocessing:

- Digital signal processing pro%ides an alternati%e method or

processing the analog signals

; First analog signals are con%ert to digital signals :

– #DC .#nalog to Digital con%erter/

#DC pro%ides the interace =et+een

the analog signal and the digital processor

; rocess the digital signals, using digital processor

; Digital processor can =e

- rogramma=le computer ,

; Small microcomputer , or

; DS chip(

; The digital output o the DS is con%erted =acG to analog

( DAC, Digital to Analog converter)


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!ample o digital processing o signals

- First, input analog signal rom microphone is con%erted to digital signal

=y #nalog to Digital Con%erter .#DC/

- Then, the digital signal is processed =y Digital Signal rocessor .DS/

- Finally, the digital output produced =y DS, is con%erted to #nalog signal,

=y Digital to #nalog con%erter .D#C/, or eeding it to speaGer

Digital

Output signal

Digital

Input signal

n is num=er(sample)

Analog

Output signal Analog

Input signal

t is time

(t) (n) ! (t)! (n)


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tages o Digital Signal rocessing o%er #nalog Signal processing:

S is more lei=le: !asily reconigura=le( .Sot+are/

(Reconfiguration of analog s!stem re"uires redesigning

#ard$are)eplication is easy, digital systems can =e easily replicated

ccuracy o design in digital systems is high

( %etting #ig# accurac! in analog s!stems is ver! difficult

&ecause of tolerances of #ard$are components)

torage o digital signals is %ery easy on magnetic media, memory, CD, DD

pen dri%e(

S allo+s implementation o more sophisticated signal processing algorithms(

.Comple mathematical algorithms can =e easily implemented/

he cost o implementation digital systems is lo+

due to the lo+er cost o the digital hard+are


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pplications o Digital Signal rocessing:

; Speech processing

; Signal transmission and reception in telephone> mo=ile systems

; Image processing

; Seismology . Study o seismic signals/

; Eil eploration .#nalysis o signals tra%eling through the layers o eart

imitations o DS:

1( The use o #DC and D#C may maGe the processing comparati%ely, slo+er (Analog s!stems are faster)

)( The po+er consumption o digital signal processors may =e higher

4( Hot suita=le or signals +ith huge =and+idth .#DC, Sampling rate has to =e t+o times the =and+idth/


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Sound signals

#udi=le range: 4 ert to 12 ert

For telephone "uality sound signals:

Range o re"uencies 0 J

thus sampling re"uency re"uired is 2 J

For CD "uality:Sampling re"uency is (1 J > channel

Range o re"uencies 0 )) J

#udi=le range o human =eings ) to ),

Hy"uist sampling theorem:

I a unction x .t / contains no re"uencies higher than B hert,

it is completely determined =y gi%ing its ordinates at a series

o points, spaced 1> .)B/ seconds apart(

.Er the signal +ith the maimum re"uency o K ert, can =e

completely represented =y sampling at a re"uency o ) K ert/


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Classiication o signals

; Signal rocessing techni"ue or any signal depends upon the

characteristics o the signal(

Multi;channel and Multidimensional Signals: 'C-'ultiple components of same signal

'D- 'ultiple inputs

Real %alued signal

s1.t/ 0 # sin 4 πt

Comple %alued signals) .t / 0 # e B4 π t 0 # cos4 πt 3 B # sin 4πt

ector representation

The signal generated =y:

multiple sources or

multiple sensors

can =e represented =y components o a %ector

s1.t/

s4.t / 0 s).t/

Three component

signal

!


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Three components o ground acceleration measured a e+ Gilometers

rom the epicenter o an earth"uaGe


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* + * + ***


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Ene dimensional> multidimensional signal

; one;dimensional signal

; I a signal is a unction o one independent %aria=le

It is one;dimensional signal

; Ether+ise it is multidimensional signal

; I it is dependent upon more than

one independent %aria=les

; #n still image can =e t+o dimensional signal

since I ., y/, intensity o light at any location

depends upon , y coordinates o the point in the picture


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; #n tele%ision picture can =e three dimensional

as I ., y, t/, intensity at any , y coordinates also

depends upon time t .rame time/

#n colour T image can ha%e three components:

Thus a colour T image is a three channel .red, green and =lue/,

three dimensional ., y and t/ signal

Ir ., y, t/

I ., y( t/ 0 Ig ., y, t/

I= ., y, t/


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lassiication o signals

( Continuous time and discrete time signals

( Continuous %alued and discrete %alued signals

( Deterministic and random signals


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Continuous;Time and Discrete Time Signals

1a( Continuous;time signals .#nalog signal/:

; # signal that eists all the time in a gi%en inter%al

; These signals are deined or e%ery %alue o time in the inter%al

; Most o the naturally occurring signals are continuous in nature

; These signal taGe on %alues in the continuous inter%al .a, =/

+here a can =e ; ∞ and = can =e 3 ∞

; Mathematically, these signal can =e descri=ed =y unctions o

a continuous %aria=le

1.t/ 0 cos π t

).t/ 0 e; L t L +here ;

∞

t ∞

(t)

t

(t)

t


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=( Discrete;time signals

The signals are deined only at speciic %alues o time

; The %alue o signal =et+een and 1, say at (A is not Gno+n

; Thus, discrete time signals eists only

at speciic %alues o time

; These time;instants usually are e"uidistant at e"ually

spaced inter%als((&ut need not &e e"uidistant)

; The discrete time signal is o=tained rom a continuous time signal

using the sampling at speciic %alues o time

* / + 0 1 2 n

(n)


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; # discrete time signal is an approimation o the continuous signal

; To impro%e the accuracy o the approimation

the sampling period .Ts / is reduced or

the re"uency o the sampling Fs is increased


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; # typical discrete time signal can =e represented =y:

.n/ 0 # cos ω(n (ω 3 * π f)

!ample o discrete;time signal:

; LtnL

.tn/ 0 e , n 0 , ± 1, ± ), ( ( ( 4 ((t), ((t), . . ((tn), 5

;I inde n o discrete;time instants is used as the independent

%aria=le .i(e(, a se"uence o num=ers/,

the signal %alue =ecomes a unction o an integer %aria=le

;Thus a discrete;time signal can =e represented mathematically

=y a se"uence o real or comple num=ers


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# discrete time signal is represented as .n/ ( instead of (t) )

; I the time instants tn are e"ually spaced, tn 0 nT .T 0 Time period/

; Signal can =e .nT/ 0 ? .T/, .1T/, ( ( .nT/ @

; There are some discrete time signals +hich are inherently discrete and

do not re"uire the sampling o continuous signal

i(e( #ccumulating a %aria=le o%er a period o time

.i(e( num=er o cars> hour/


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.n/ 0 (2n or n N and

.n/ 0 or n (negative values of n)

.n/

n

*raphical representation o the discrete time signal


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( Continuous;%alued and Discrete;alued signals

; The %alues o continuous time or discrete time signals

can =e:

continuous (ta6ing all possi&le values) or

discrete ( ta6ing onl! a finite set of values)

)a( Continuous;alued signal:

Signal taGes on all possi=le %alues ina range . +hich can =e inite or an ininite/

.y;ais can =e di%ided into ininite num=er o le%els/

. (A, (A), (A4, (6 ( ( (/


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)=( Discrete;alued signal:

Signal taGes on %alues only rom a inite set o possi=le %alues

(!-ais can &e divided into finite num&er of levels)

; Osually, these %alues are e"uidistant,

hence can =e epressed as an integer multiple o

distance =et+een t+o successi%e %alues(

.), 4, , A or 6, not )(4)), 4(1, (4A, 6(1)/

Digital signal:


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Digital signal:

# discrete;time signal ha%ing a set o discrete %alues(

For digital processing o a signal:

; It must =e discrete in time

; Its %alues also must =e discrete

; Digital signal is o=tained =y:

First - E=taining a discrete time signal =y

sampling an analog signal at discrete instants in time

Then – 'uantiing the %alues o discrete time toa set o discrete %alues

'uantiation:

It is the process o con%erting continuous;%alued signal into

discrete %alued signal =y simple rounding > truncation process

or =y mapping to a set o inite %alues


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Digital signal +ith dierent amplitude %alues: .1, ), 4 and /

1, 1, 1, ), 1, 4, ), , ), 1<

-* - * / + 0 1 n

7(n)


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Discrete time signal

It is deined or e%ery integer %alue o n or ;∞

n 3∞

.n/ is nth sample o the signal

i .n/ is o=tained =y sampling then . n/ ≡ .nT/

4 (8, (8), . . ,(n8)

+here T is sample period

i(e( the time =et+een t+o successi%e samples

#lternati%e representation o DT signals:to graphical representation

1( Functional representation,

1, or n 0 1, 4

.n/ 0 , or n0)

, other+ise

)( Ta=ular representation:

n ( ( ( ;) ;1 1 ) 4 A 6 ( ( (

;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

.n/ 1 1


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( Se"uential representation:

#n ininite duration signal or

se"uential +ith the time origin .n 0 /

indicated =y sym=ol↑

.n/ 0 ( ( ( , , 1, , 1, , , ( ( ( <

↑

# se"uence +hich is or n0 .n/ 0 , 1, , 1, , , ( ( ( <

↑

The time origin or a se"uence +hich is ero or n

; First letmost point is considered to =e the origin

inite duration se"uence

.n/ 0 4, ;1, ;), A, , , ;1< .se%en point se"uence/ ↑

# se"uence .n/ 0 or n .n/ 0 ,1, , 1< .;point se"uence

↑


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Some elementary Discrete;time Signals

1( The unit impulse se"uence

1, or n 0

Denoted as .n/ , or n

≠

; It is ero e%ery+here ecept n0,

+here it has a unit height

; #lso reerred as a unit impulse

. ero e%ery+here, ecept time t 0 /

; It has unit area

.n/ ≡

-* - * / +

)( The unit step signal

Denoted as u .n/ 1, or n ≥

, or n

; It is 1 at positi%e n . including at n0 / and

at negati%e n

u .n/≡

-* - * / +

%rap#ical representation

of unit impulse

%rap#ical representation

of step signal


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4( The unit ramp signal

denoted as u r .n / n, or n ≥

, or n u .n/ ≡

-* - * / +

( The eponential signal

; a se"uence o the orm .n/ 0 an

or all n 9 a 9

a


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;nit step signal

;nit ramp signal


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.n/ 0 a n

. /


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.n/ 0 a n


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I the parameter PaQ is real .n/ is a real signal

; i the parameter PaQ is comple %alued,

a≡

r e Bθ +here r andθ

are the parameters

.n / 0 r n e B θn

0 r n . cos θn 3 B sin θn/

Real part g .n/ 0 rn

cosθ

n

Imaginary part B .n/ 0 rn sin θn

I r 0 ( and θ 0 π >1

g .n/ 0 rn cos θn 0 .(/n cos π >1 ( H

i .n/ 0 rn sin θn 0 .(/n sin π >1 ( H

g .n/ and B .n/ are a damped, decaying . eponential/ cosine unctionand dam ed sine unction

.n/ 0 a n

.n/ 0 a n :

Real part


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Real part g

Imaginary part


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Imaginary part i


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.n / 0 r n e B θn can =e represented as amplitude and phase unction:

#mplitude unction L .n/L 0 # .n/ ≡ r n

hase unction∠

.n/ 0φ

.n / ≡

θ

n (π n


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Classiication o Discrete;time signals:

1( !%en and odd signals .Symmetric and asymmetric signals/

)( eriodic signals and aperiodic signals

4( !nergy signal and po+er signal


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1( !%en and odd signals .Symmetric and asymmetric signals/

# signal is e%en > or symmetric i .n/ 0 .;n/

# signal is odd > or asymmetric i .;n/ 0 ; .n/ or

.n/ 0 ; .;n/

For an e%en signal .1/ 0 .;1/, .)/ 0 .;)/ ( ( (

For an odd signal .1/ 0 ; .;1/, .)/ 0 ; .;)/ ( ( (

; I .n/ is odd then . / 0

!amples:

# sine +a%e is odd,

+hile a cosine +a%e is e%en signal

sin .A/ 0 ; sin.;A/

cos .A/ 0 cos . ;A/


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-+ -/ -* - * / + n e%en signal

* / + n odd signal

-+ -/ -* -

(n)

(n)

.n/ 0 .;n/

.n/ 0 ; .;n/

. / 0

(n)


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#ny ar=itrary signal can =e epressed as the sum o the t+o components:

; Ene e%en and the

; Ether odd

; The e%en signal component can =e ound out =y:

e.n/ 0 ?.n / 3 .;n/@

odd signal component

o .n/ 0 ? .n/ – .;n/@

#dding =oth:

.n/ 0 e.n/ 3 o.n/

Fi d d dd t th i di t i l


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Find e%en and odd components o the gi%en discrete signal:

.n/ 0 1, ), 4, , 1, ), )<

↑

.;n/ 0 ), ), 1, , 4 ), 1< mirror image o .n/

↑

a=out origin ./

.n/e 0 1(A, ), ), , ), ), 1(A< 0 ? .n/ 3 .;n/@ > )

↑

.n/o 0 ;(A, , 1, , ;1, , (A< 0 ? .n/ ; .;n/@ > )

↑


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)( eriodic signals and aperiodic signals

; # signal .n/ is periodic +ith period H .HN /

Enly i .n 3H / 0 .n/ or all %alues o n

Smallest %alue o H or +hich the a=o%e e"uation hold good

is called the undamental period

;The signal is non;periodic i there is no %alue o H

that satisies the a=o%e e"uation


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ChecG i .n/ 0 # cos .ωn 3 θ / is periodic

ereω

0 )π

.n/ 0 # cos .)π n 3 θ /

.n 3H/ 0 # cos .)π

.n 3H/ 3θ

/0 # cos .)π n 3 )π H 3 θ /

For periodicity:

.n 3 H/ 0 .n/

# cos .)π

n 3 )π

H 3θ

/ 0 # cos .)π

n 3θ

/

For this to =e true:

)π H 0 0 or 0 )πG +here G is an integer (, , *, / . . )

or 0 G > H +here =oth G and H are integers

Thus or periodicity, 0 G >H, +here =oth G and H should =e integers


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1( ChecG i cos.(1 πn/ is periodic8

(1π

0 )π

0 (1> ) 0 1>) cycles per sample,

0 G> H ratio o t+o integers

#s G 01 and H is ), =oth are integers, so cos .(1 πn/ is periodic

)( ChecG i .n/ 0 sin 4nere 4 0 )π or 0 4> ) π

since cannot =e epressed as raction o t+o integers ,

the signal .n/ 0 sin4n is not periodic

nergy and po+er signal


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nergy and po+er signal

!nergy o a signal or discrete time signal .n/

∞

is deined as ! ≡ L .n/ L )

n 0 ;∞ #s the magnitude s"uare is used or .n/,

The deinition is applies to real as +ell as comple;%alued signals

The energy o a signal can =e inite or ininite

!nergy signal

; I !nergy ! is inite then .n / is called an energy signal

; The inite !nergy can =e called ! o signal .n/

Many signals possess ininite energy,=ut ha%e a inite a%erage po+er

1 H

%erage o+er 0 lim ∑ L .n/L )

H → ∞ H 3 1 n 0 ; H


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! d it t i l . /


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!nergy and po+er o a unit step signal u.n/

P1Q or n 0 ≥

; Step signal u.n/ 0

PQ other+ise

∞ ∞

!nergy ! ≡ L .n/ L ) 0 Lu .n/ L ) 3 = = . . . 3 ∞

n 0 ; ∞

Since the ! is ininite, the unit step signal is not an energy signal

1 H

The a%erage o+er 0 lim ;;;;;;;;;;; ∑ u ) .n/

H → ∞ )H 31 n 0

( summation of u* (n) is > =)

1 .H31/ 1 3 1> H 1

0 lim ;;;;;;;;;;; 0 lim ;;;;;;;;;;; 0 ;;;;;;;

H → ∞ )H 31 H → ∞ ) 3 1>H )

Conse"uently, the unit step se"uence is po+er signal

and its energy is ininite

* / . .

Find !nergy or the signal .n/ 0 an u .n/ aL 1 ?u (n) step signal5


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Find !nergy or the signal .n/ 0 an u .n/ aL 1 ?u (n) step signal5

∝

; #s energy o D( T( Signal is: ! H ≡ ∑ L .n/L)

n 0 ;∝

0 ∑ Lan u .n/L) for n 0 to ∝ u (n) is a init step

Since u .n/ is 1, or to ∞ 0 ∑ Lan ( 1L) or n 0 to ∝

0?a)@ 3 ?a)@1 3 ?a)@) 3 . .

.*eometric series ∑ #n 0 1 3 # 3 #) 3 #4 ( ( ( 0 1> 1 –# i # 1/

Thus ! 0 1> 1; a) i La)L 1

! l i h th th ll i i l i l i l


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; First calculate energy o the signal

! o .n/ is gi%en =y:

n 0 ∞ n 0 ∞

! 0 L .n/L ) 0 L (An u .n/L )

n 0 ;∞

n 0 ;∞

-Since this signal is multiplied =y unit step: n is rom ; ∞

( signal is ero for n )

n 0∞

n 0∞

n 0∞! 0 L .1>)/ n L ) 0 .1>)/ ) n 0

.1>/ n

n 0 n 0 n0 n 3 ∞

?@tandard %eometric series Σ A n = A = A*= A / . . . < (-A) .# 0 /<

n 3 8#us ! 0 1 > .1 ;1> / 0 1> U 0 > 4 0 1(444

- Thus ! is inite

Since !nergy is inite it is energy signal

!plain +hether the ollo+ing signals are po+er signal or energy signal(

i (A n u .n/


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Eperations on discrete time signals:

The mathematical transormation rom one signal to another

is represented as:y .n/ 0 T ? .n/@

; Eperations in%ol%ing:

; Independent %aria=le . time/

; Dependent %aria=le .amplitude/

The =asic operations on DT signals are:

1( Time shiting

)( Time re%ersal Independent varia&le, time

4( Time scaling

( Scalar multiplication

A( Signal addition and

multiplication

Dependent varia&le,

Amplitude


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Eperations in%ol%ing independent %aria=le . time/

1( Time Shiting

; Shit n =y ; G in .n/ Delay.G is an integer/

y .n/ 0 .n ; G/ for 6 3 /, ! () 3 (-/)

; i G is negati%e the shiting results in ad%ancing =y LGL units

- .n / 0 ;1, , 1, ), 4 , , , , , <

↑

-.n;4/ 0 ;1, , 1, ), 4 , , , , , < shiting delay G 0 4 (-/ &ecomes origin)

↑

-. n 3)/ ;1, , 1, ), 4 , , , , , < ad%ancing G0 ;) (=* &ecomes origin)

↑

Hote: delay >ad%ance is easy in stored signal

.o+e%er, ad%ancing in real time, generated signal is not possi=le(/

Future

)( Time re%ersal –


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Folding> Relection a=out the time origin n 0 :

; Replacing n =y –n y .n/ 0 .;n/

.n / 0 ), ), ), , 1, ), 4, <

↑ .;n/ 0 , 4, ), 1, , ), ), )<

↑

.;n ; )/ 0 , 4, ), 1, , ), ), )< .Folding and delayed =y )/

↑ .;) =ecomes origin/

4( Time scaling:


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g

; Changing in time scale

; T+o types do+n scaling and up scaling

; Do+n scaling is represented as:

y .n/ 0 T ? .n/@

!ample y .n/ 0 .)n/

.n/ 0 ;4, ;), ;1, , 1, ), 4, , , , , , , <

↑

y .n/ is taGing e%ery other sample rom .n/

y./ 0 ./ 0 , y.;1/ 0.;)/0 )

y .1/ 0 .)/ 0 y.;)/ 0.;/0

y.)/ 0 ./ 0 ,

Thus y .n/ ;), , ) , , , , <

↑

.n/ 0 ;4, ;), ;1, , 1, ), 4, , , , , , , <

↑

y .n/ 0 .)n/ 0 ;), , ), , , , <

↑

Sampling is reduced to hal


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Op scaling


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Op scaling

y .n/ 0 .n > ) /

.n/ 0 A, , 4, ), 1, ), 4, , A<

↑

y . /0 .>)/ 0 ./ 0 1

y .1/0 .1>)/ 0 .(A/ 0 no sample

y .) /0 .)>)/ 0 .1/ 0 )

y .4 /0 .4>)/ 0 .1(A/ 0 no sample

y . /0 .>)/ 0 .)/ 0 4

y .6 /0 .6>)/ 0 .4/ 0 y .2 /0 .2>)/ 0 ./ 0 A

Thus y .n/ is epanded %ersion o .n/ +ith y .1/, y.4/, y.A/ ( ( Ho sample

- -1 - -0 -+ -/ -* - * / + 0 1

8#e sampling rate is increased from


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#ddition, multiplication, and scaling o se"uences

; #mplitude modiication includes addition, multiplication and scaling

o discrete;time signals

( #mplitude scaling:

; Multiplying =y a constant

y .n/ 0 # .n/ ; ∞ n ∞

A( Signal addition and multiplication

; The sum o t+o signals

y .n/ 0 1 .n/ 3 ) .n/ ; ∞ n ∞

.add corresponding terms)

; The product o t+o signals

y .n/ 0 1 .n/ ( ) .n/ ; ∞ n ∞

.multiply corresponding terms)


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At -*, - . 3

-, * . 3

, * . 3 *

, . * 3 *

*, . / 3

Discrete time Systems


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Sy

; DT system is a de%ice> algorithm that operates on

a discrete;time signal .input> ecitation/,according to some +ell;deined rule,

to produce another discrete;time signal .output> response/

; It is a set o operations perormed on input signal .n/ to produce

output signal y .n/

.n/ is said to =e transormed to y .n/

y .n/ ≡ T ? .n/ @ +here T is transormation

or .n /→ y .n/ .transormed to /

Determining the response:

LnL 4 n 4


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LnL, ; 4≤

n≤

4

.n/ 0 , other+ise? .n/0 , 4, ), 1, , 1 ,), 4, <

↑

V .n/ 0 .n ;1/ 0 , 4, ), 1, , 1 ,), 4, 4, ), 1, )>4 , 1 ,), A>6, 1, , <

↑

(n ) 3 ? , , /, *, , , , *, /, , ,

B() 3


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y . / . . / . / . /

.n/0 , 4, ), 1, , 1 ,), 4, <

↑

0 , 4, 4, 4, ), 1, ), 4, 4, 4, <

↑

n

y .n/ 0∑

.G/ 0 .n/ 3 .n;1/ 3 . n;)/ 3 ( ( B ( ) 3 () = (-) = (-*)

G 0 ; ∞ = (-/), = (-+) = (-0)

.n/0 , 4, ), 1, , 1 ,), 4, <

↑

y .n/ 0 , 4, A, 6, 6, 5, ,1),1)< . G 0 ; ∞/ sum o all past %alues 8

↑

y./ 0 ./ 3 .;1/ 3 .;)/ ( (

= /

= / =*

= / =*=

= / =*== =

Classiication o Discrete Time Systems


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y

Discrete time system is a de%ice or algorithm that operates on

a discrete time signal

; It is represented as y .n/ 0 T ? .n/@

1( Static and Dynamic systems (@tatic - current input)

)( Causal and #nti;causal systems .Causal - resent and past

no future

4( $inear and Hon;linear systems

( Time %ariant and Time in%ariant systems

A( Sta=le and unsta=le systems

6( FIR and IIR systems

(Finite Impulse Response< infinite impulse response)

Static %ersus Dynamic systems


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y y

Static system:

# system is static i the output o a system depends

only the current input andnot on the past or uture input

; !amples o static systems:

y .n/ 0 T ? .n/@

y .n/ 0 ( .n/, y .n/ 0 a .n/

y .n/ 0 log .n/

y .n/ 0 # cos .n/

y .n / 0 a .n / 3 = 4 .n/

; In each case y .n/ re"uires only present %alue o input .n/

; Static systems are also called memory less system

as no memory is re"uired to store pre%ious input %alues

e( in case o y .n/ 0 ( .n/, ! () 3 + ( (), ! (*) 3 + ( (*), . . .

Dynamic system


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; #ll the systems that are not static(

; In dynamic systems, the output may depends on the

present as +ell as

past input signals (ven future signals)

y .n/ 0 .n/ 3 .n;)/ is a dynamic system

; as or inding y ./ 0 ./ 3 .)/ . current input = past input)

past input is re"uired

; Thus the dynamic system re"uires memory to store the past inputs

Dynamic system is also called memory system

!amples:


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p

y .n/ 0 ? .n/ 3 .n;1)/@

for ! (0), (0) and (/) are re"uired

y .n/ 0 .n/ ( .n;1/

n

y .n/ 0 ∑ .n ;G/ Re"uires Finite memory

G0

ny .n/ 0 ∑ .n ; G/ Re"uires Ininite memory

G0 ; ∞

)( Causal and Hon;causal system


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; # system is causal i the output o the system y .n/ at any n

depends on present and past inputs =ut

not on uture inputs

; !ample:

y .n / 0 .n/ 3 .n;)/ 3 .n;4/

.present past past)

Hon;causal system:


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; I the output also depends upon the uture input

then it is Hon causal system

; !ample:y .n/ 0 .n/ 3 .n31 / Future

(present = future) y ./ 0 ./ 3 .A/

; ence or a non;causal system,

; Future input needs to =e predicted to ind the present

Ether eamples:

y .n/ 0 .)n/

y .n/ 0 .n/ – .n3)/

4( $inear and non; linear


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; $inear systems satisies superposition principle:

; The superposition principle states that the response to

a +eighted sum o input signals,

should =e e"ual to the corresponding

+eighted sum o the outputs o the system

to indi%idual input signals(

i( e(T? a11.n/ 3 a)).n/ @ 0 a1T ? 1.n/ @ 3 ? a) T ? ).n/ @

(response to t#e $eig#ted sum of inputs 3

t#e $eig#ted sum of responses to t#e individual inputs)


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1 .n/

) .n/

3 T y1 .n/ 0 T ?a1 1 .n/ 3 a) ) .n/@a1

a)

1 .n/

) .n/#HD

T

T 3 y) .n/ 0 a1 T ?1 .n/@ 3 a) T ? ) .n/@

a1

a)

ChecG +hether the y .n/ 0 n( .n/ is a linear or non linear system8


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#s y .n/ 0 T ? .n/@

ence, the system is n ( .n/

For linearity:

ChecG i T ?1 .n/ 3 ) .n/@ 0 T ? 1.n/ @ 3 T?) .n/@ or

! (n) 3 !* (n)

so y1

.n/ 0 T ?1

.n/ 3 )

.n/@

0 n( ?1 .n/ 3 ) .n/@ 0 n( 1 .n/ 3 n() .n/

y) .n/ 3 T ? 1.n/ @ 3 T?) .n/@ 0 n ( 1.n/ 3 n( ) .n/

ence y1 .n/ 0 y) .n/

The system y .n/ 0 n( .n/ is linear

similarly y .n/ 0 n ( ) .n/ is non;linear can =e pro%ed

Time;%ariant and time; in%ariant systems


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Time;in%ariant system:

In Time;in%ariant system

the input >output characteristic does not change +ith time

; suppose input .n / produces y .n/ at any stage,

y .n/ 0 T ? .n/@

; i the input signal is delayed =y G units,

the output y .n; G/ +ill =e same as y .n/ ecept that it is delayed =y G units

- First delay the input =y G samples , and o=ser%e the output → y .n, G/- Delay the output =y G samples → y .n ; G/

I y .n; G/ 0 y .n, G/ the system is time;in%ariant


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.n ;G/ y .n ;G/ .n/ y .n/T T

ChecG +hether the ollo+ing systems are time in%ariant8


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1( y .n/ 0 e .n/

; Delay input =y PGQ y .n, G/ 0 e .n; G/

; Delay the output y .n/ =y G, y .n ; G / 0 e .n;G/

Since =oth are e"ual the system is time;in%ariant

)( y .n/ 0 n ( .n/ is time %ariant as

1( Delaying input y .n, G/ 0 n( .n ; G/

)( Delaying output y .n ; G/

y .n; G/ 0 . n; G/ ( .n; G/

Koth are not e"ual the system is time;%ariant

Sta=le and unsta=le systems :

For a Sta=le system


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For a Sta=le system

a =ounded input, produces =ounded output

$et M is inite num=ers such that M ∞

,the input is said to =e =ounded i L .n /L ≤ M ∞

for all value of n

My is a inite num=er such that My ∞,

the output is said to =e =ounded i Ly .n/L≤

My ∞

for all value of n

; I or a =ounded input .n/, there is ∞ output

the system is unsta=le

ChecG +hether the ollo+ing systems are sta=le8

1( y .n/ 0 e .n/

)( y .n/ 0 .)n/

Koth are sta=le, as or =ounded input, there is =ounded output

#nalysis o $inear Time –In%ariant systems .$T I / systems:


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; $T I systems are characteried in time domain =y

their response to a unit sample se"uence

; #ny ar=itrary signal can =e represented as

+eighted sum o unit sample se"uences

Techni"ues or analysis o $inear systems:

1( Direct solution o the input;output e"uation

)( First decomposing the input signals into elementary signals, then

determining the response to each elementary signal and

adding all the responses

Direct solution o the input;output e"uation

I t t t ti h th


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Input –output e"uation has the orm

y .n/ 0 F ? y .n;1/, y .n;)/, ( ( V .n;H/, .n/, .n;1/, .n;1/, . . . - .n;M/@

F ?( ( ( ( @ denotes some unction o "uantities in =racGets

; Specially, or $TI .$inear Time;in%ariant/ systems

; The general orm o input;output relationship:

H M

y .n/ 0 ; ∑ aG y .n ; G/ 3 ∑ =G .n ; G/

G01 G 0aG and =G are constant parameters that speciy the system and

aG and =G are independent o .n / and y .n/

- The a=o%e e"uation is called a dierence e"uation- It represents one +ay to characterie the =eha%ior o a

discrete;time $TI system

Second method o analying the =eha%ior o a linear system


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; Resol%e the input signal .n/ into

+eighted sum o elementary signal components ?G .n/ @

- (8#e elementar! signals are selected so t#at t#e response

to eac# component is easil! determined)

- Ose the linearity property to add the responses to

indi%idual components to o=tain the total response .output/

.n / 0 ∑ cG G .n/ ()

G

G .n/ are the elementary signal component

cG are the set o amplitudes .+eighting coeicients/

in the decomposition o the signal .n/

I the y .n/ is the response o the system to


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I the yG .n/ is the response o the system to

the elementary component G .n/

then yG .n/≡

T ?G .n/@ (*)

#ssuming the system response to the cG G .n/ is cG ? yG .n/@

#s a conse"uence o scaling property o the linear systems (n) from ()

y .n/ 0 T ? .n/@ 0T ?∑

cG G .n/ @ 0∑

cG T ? G .n/@G G

Osing additi%e property o the linear system

y .n/ 0 ∑ cG yG .n/ ( ! 6 (n) from (*)

G

(8#e response to t#e input (n ) $#ose components are c6 (n)

e"uals to t#e $eig#ted sum of

t#e responses to t#e components)

Resolution o a discrete time signal into impulses


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- $et an ar=itrary signal =e .n/ ? , *, /, , *, , /

↑

; Resol%ing the signal .n/ into a sum o unit sample se"uences

let Gth component o .n/, G .n/ 0 .n ;G/ 4 (n- 6)t# impulse5

+here G is the delay o the unit sample se"uence

; Multiplying .n/ and .n ;G/ 6 3/

Since .n ; G/ is ero e%ery+here ecept at n 0 G .Impulse/

the result o the multiplication +ill =e a se"uence:

+hich is ero at e%ery+here ecept at n 0 G

Thereore .n/( .n; G/ 3 .G/( .n ; G/

-/ -* - * /

(n)

-/ -* - * /

(/)

i +e taGe dierent %alues o G +e +ill get:


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i +e taGe dierent %alues o G +e +ill get:

∞

.n/ 0∑

.G/ .n ;G/

G 0 ; ∞

0 summation o an ininite num=er o unit sample se"uence , .n ;G/

ha%ing amplitude %alues o .G/

Representation o a signal in terms o

+eighted sum o shited, discrete impulses :


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+eighted sum o shited, discrete impulses :

#ny ar=itrary signal can =e represented as

summation o shited and scaled impulses

.n/ 0 (A, ), 1, (A, 1, ), (A<

↑

; The sample ./ can =e o=tained =y

multiplying ./, the magnitude, +ith a unit unction .n/

./ W or n 0i( e( ./ ( .n/ 0 0 (A ( 1 0 (A . multipl!)

W or n≠

δ (n)

n n

() . δ (n) .0

-/ -* - * /

.n/ 0 (A, ), 1, (A, 1, ), (A


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↑

Similarly other components can =e o=tained:

:

.1/( .n;1/ 0 1 ( 1 0 1

.)/( .n;)/ 0 ) ( 1 0 )

.4/( .n;4/ 0 (A ( 1 0 (A

.;4/( .n 34/ 0 (A ( 1 0 (A

.;)/( .n 3)/ 0 ) ( 1 0 )

.;1/( .n 31/ 0 1 ( 1 0 1

Thus .n/ 0 (A( .n 34/ 3 ) ( .n 3)/ 3 1. .n 31/ 3

3 (A ( .n/ . 3 1 ( .n ;1 / 3 ) ( .n ;)/ 3 (A ( .n ;4/

The signal .n/ can =e %ie+ed as a summation o scaled andshited impulses

∞

in general, .n/ 0∑

.G/( .n ;G/

6 3 - ∞

$et .n / 0 ) 4


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$et .n / 0 ), , , 4<

↑

Resol%e into a sum o +eighted impulse se"uence

since .n / is non;ero at the time instant n 0 ;1, , )

+e need three impulses at G 0 ;1, , )

.n/ 0 ) (

.n31/ 3 ( .n/ 3 4 (

.n;)/

↑

Impulse response and con%olution

. / T ? . / @ O t t ( ) t f f ti ( )


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y .n/ 0 T ? .n/ @ Output ! (n) 3 transfer function (n)

I input .n/ is a unit impulse .n/ then the output o the system is

Gno+n as impulse response h .n/

; Thereore impulse response: h .n/ 0 T ? .n/@

.transfer function of unit impulse)

Impulse response completely characterie the system

.n/ y .n/8

.n/ h .n/8

Con%olution sum


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Since input signal .n/ can =e represented as

+eighted sum o discrete impulses

∞ i( e( .n/ 0 ∑ .G/( .n; G/ ($#ere n is t#e time inde,

G 0 ; ∞ 6 is a parameter s#o$ing

t#e location of input impulse

∞

Thus output signal y .n/ 0 T ? .n/@ 0 T ∑ .G/( .n ;G/

G 0 ;∞

; I the system is linear :

∞

y .n/ 0 ∑ T ? .G/( .n; G/ 5

G 0 ;∞

∞

y .n/ 0∑

.G/( T ? .n; G/ 5 () G 0 ;

∞

$et T ? .n G/@ 0 h .n G/ # (n 6) unit pulse response dela!ed


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$et T ? .n ;G/@ 0 h .n, G/ # (n, 6) unit pulse response dela!ed

∞

y .n/ 0 ∑ .G/( h .n, G/ (From ()

G 0 ;∞

I the system is time in%ariant: h .n, G/ 0 h .n ;G/ ( Response $it#

t#e input dela!ed 3

8#e response $it#

∞

t#e output dela!ed)

y .n/ 0∑

.G/ ( h .n ; G/

G 0 ;∞

. The output> response at PnQ is summation o

%alues o input at G, multiplied =y unit impulse response at n ; G,

or all %alues o G/

∞

y .n/ 0 ∑ .G/( h .n ; G/


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y . / . / . /

G 0 ;∞

The a=o%e e"uation is called con%olution sum:

For a $inear Time In%ariant .$TI/ system

i input se"uence .n/ and

the impulse response h .n/ is Gno+n

The response y .n/ can =e ound out =y the con%olution sum

∞

y .n/ 0 ∑ .G/( h .n ; G/

G 0 ;∞

(Output ! at n 3 sum of values of input (n) at n 3 6,multiplied &! unit impulse response at n -6,

for all values of 6

The con%olution sum is represented as

y .n/ 0 .n/ X h .n/

X

roperties o Con%olution


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1( Commutati%e $a+ : .n/ X h .n/ 0 h .n/ X .n/

G Gy .n/ 0 ∑ .G/( h .n ; G/ 0 ∑ h . G/( .n ;G/(

G 0 ; ∞ G 0 ; ∞

)( #ssociati%e $a+: ? .n/ X h1 .n/ @ X h) .n/ 3 .n/ X ? h) .n/ X h1 .n/ @

4( Distri=uted $a+: .n/ X ? h1 .n/ @ 3 h) .n/@ 3 .n/ X h1 .n/ 3 .n/ X h) .n/

Computation o $inear Con%olution


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1( *raphical Method

)( Ta=ular method

The linear con%olution is gi%en =y:

∞

y .n/ 0 ∑ .G/ ( h .n ; G/

G 0 ; ∞

( Response ! (n) at n, is summation of values of (n) at 6,multiplied &! unit impulse response at n - 6, for all values of 6)

Calculating the %alue y .n/ or time instant n 0

∞ ∞

y./ 0∑

.G/( h. ; G/ 0∑

.G/( h. ; G/ G 0 ; ∞ G 0 ; ∞

. h . ;G/ indicates olding /

For n 0 1


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∞

∞

y.1/ 0 ∑ .G/( h.1 ; G/ 0 ∑ .G/( h.; G 3 1/

G 0 ; ∞ G 0 ; ∞

h.; G 3 1/ indicates shiting o olded signal h

h .;G 31/ Indicates h .;G/, the olded signal is delayed =y P1Q sample

For n 0 )

∞ ∞

y.)/ 0∑

.G/( h.1 ; G/ 0∑

.G/( h.; G 3 )/ G 0 ; ∞

G 0 ; ∞

h.; G 3 )/ indicates shiting o olded signal h

h .;G 3)/ Indicates h .;G/, the olded signal is delayed =y P)Q sample

#nd so on

Steps or computing the con%olution =et+een .G/ and h .G/


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1 Folding: old h .G / # ( -6)

) Shiting: shit h . ;G/ # (-6 =)

4 Multiplication: Multiply .G/ =y h .n –G/,

ind all the product terms or all the %alues o G

Summation: Sum all the product terms to ind y .n/

Find y .n/ or all %alues o n, ollo+ing steps ) to or each

%alue o n

∞

Range o PnQ and PGQ , or calculating y .n/ 0∑ .G/ ( h .n ; G/


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g g y . / . / . /

G 0 ; ∞

Range o PnQ $o+est %alue o n in y. n/ 0 lo+est %alue o n in .n/ 3 lo+est %alue n in h h .n/

to

ighest %alue o n in y. n/ 0 highest %alue o n in .n/ 3

highest %alue n in h .n/

Range o PGQ

The range G +ill =e same as range o n in .n/

E=tain the impulse response o a system h .n/ 0 1, ), 1, ;1<

↑

I t Si l . / 1 ) 4 1


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Input Signal .n/ 0 1, ), 4, 1<

↑

Range o n in y .n/ 0 .$o+est %alue o n in .n/ 3 lo+est %alue o n in h .n//

to .highest %alue o n in .n/ 3 highest %alue o n in h .n//

n in y .n/ 0 .;1 3 0 ;1/ to .) 3 4 0 A / 0 ;1 to 3 A

Range o G 0 range o n in .n / 0 to 4

Determining the response> output y: For n 0 : y ./ h .n/ 0 1, ), 1, ;1


99/119

y . / . / . / , , , <

y .1/ 0∑

.G/ ( h .;G 31/ G %aries rom to 4↑

.G/ ( h .;G 31 /

y .1/ 0∑

1, ), 4, 1< ( ;1, 1, ), 1


100/119

Similarly y ./ 0 ;)

y. A/ 0 ;1

y . ;1/ 0 1

!ntire response o the system ( ( ( (, , , 1, , 2, 2, 4, ;), ;1, , , ( ( ( < ↑

.G/ ( h .;G 34 /

y .4/ 0 ∑ 1, ), 4, 1< ( , ;1, 1, ), 1


101/119

.n/ 0 1, ), 1, )< h .n/ 0 1, 1, 1<

Since the se"uences are gi%en in terms o PnQ

o=tain .G/ and h .G / =y su=stituting =y G

.G/ 0 1, ), 1, )< h .G/ 0 1, 1, 1<

↑

↑

Con%olution is gi%en =y:

∞

y .n/ 0∑

.G/( h .n ; G/ G 0 ;

∞

Range o PnQ or y .n/ 0 $ 3 h$ 0 3 0 to h 3 hh 0 ) 3 4 0 A

0 to A

Range o PGQ : same as the %alue o n in .n/ 0 to 4

Ta=ulation method o linear con%olution


102/119

$et .n/ 0 ./, .1/, .)/ < and h .n/ 0 h./, h.1/, h.)/<

↑ ↑

Step I: Form the matri as sho+n =elo+:.n/

./ .1/ .)/

h./

h .n/ 0 h.1/

h.)/

Step II Multiply the corresponding elements o .n/ and h .n/ and


103/119

Step II Multiply the corresponding elements o .n/ and h .n/, and

Step III Separate out elements diagonally as sho+n =elo+:

.n/

./ .1/ .)/h./ h./(./ h./(.1/ h./(.)/

h.n/ 0 h.1/ h.1/(./ h.1/(.1/ h.1/(.)/

h.)/ h.)/(./ h.)/(.1/ h.)/(.)/

Step I : #dd the elements in each =locG(

Thi ill i di l . /


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This +ill gi%e corresponding %alues o y .n/

V./ 0 h./(./

V.1/ 0 h./(.1/ 3 h.1/(./ V.)/ 0 h./(.)/ 3 h.1/(.1/ 3 h.)/(./

V.4/ 0 h.1/(.)/ 3 h.)/(.1/

V./ 0 h.)/(.)/

V .n/ 0 y.1/, y.)/, y.4/, y./<

Range o n in y .n/ 0 $o+est %alue o n in .n/ 3 lo+est %alue o n in h .n/

to highest %alue o n in .n/ 3 highest %alue o n in h .n/

0 . 3 0 / to .) 3 ) 0 / 0 to 3

The range o PnQ or y .n/ 0 to

Compute con%olution y .n/ 0 .n/X h. n/

. / 1 1 1 1< h . / 1 ) 4


105/119

.n/ 0 1, 1, , 1, 1< h .n/ 0 1( ;), ;4, <

↑

↑

.n/ 1 1 1 1

h .n/1 1 1 1 1 (

;) ;) ;) ;) ;) ( -*

;4 ;4 ;4 ;4 ;4 .-/

. +

y .n/ 0 1, 1;), ;);4, 13 ;43, 1 ;) 3 3, ;) ;4 3, ;43 , <

0 1, ;1, ;A, ), 4, ;A, 1, <

∑ .n/ ( ∑ h(.n/ 0 ∑ y .n/

.13133131/ ( .1 ;) ;4 3/ 0 . 1 ;1 ;A 3)34 ;A 31 3/

( 0

Erigin

Correlation:

; Correlation is used or the comparison o t+o signals

It i d t hi h t i l i il


106/119

; It is measure o degree to +hich t+o signals are similar

; Cross correlation

Correlation o t+o separate signals is Gno+n ascross correlation

; #uto correlation

Correlation o a signal +ith itsel is Gno+n as

auto correlationCross Correlation:

Correlation =et+een t+o signals .n/ and y .n/

3 ∞

r y . l / 0∑

.n/ ( y .n ; l/ +here l 0 ,±

1,±

), ±

4 ( ( (n 0 ; ∞

Inde l is time shit .or lag/ parameter and E

the su=scripts , y on cross correlation se"uence r y . l /

indicates the se"uences =eing correlated


107/119

#uto correlation

The correlation o a signal +ith itsel to determine time delay


108/119

The correlation o a signal +ith itsel to determine time delay

=et+een the transmitted signal and the recei%ed signal

3∞

r .l/ 0∑

.n/ ( .n ; l/ +here l 0 ,±

1,±

), ±

4 ( ( (n 0 ; ∞

Compute the cross;correlation =et+een:

.n/ 0 1 1 1


109/119

.n/ 0 1, 1, , 1<

↑

y .n/ 0 , ;4, ;), 1<

↑

; Cross correlation o .n/ and y .n/ is gi%en =y :

3 ∞

r y.l/ 0 ∑ .n/ ( y .n ; l/ +here l 0 , ±1, ±), ±4 ( ( (

n 0 ; ∞

; Range o n:#s y .n/ is delayed =y l, and .n/ is not changed

so the range o n in the summation is

same as .n/ 0 ;) to 31

1

So r y .l/ 0 ∑ .n/ ( y .n ; l/

n 0 ;)

Range o n in .n/ is ;) to 31

Range o n in y .n/ is ;) to 31


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Range o l:

since the e"uation y .n; I/ should ha%e maimum %alue at y .n/ 0 y.1/

n – l 0 1 maimum

Starting %alue o PnQ in summation is ;) in (n)

;) –l 0 1 or n 0 ;)

I 0 ;1 ;) 0 ;4

or starting %alue o l 0 ;4

The e"uation y .n – l / should ha%e minimum %alue at y n / 0 y .;)/

n; I 0 ;)

=ut the summation should stop at n 01 in (n)

1; I 0 ;)

or I 0 4

So the range o I 0 ;4 to 3 4

The range o n 0 ;) to 31

The range or l is ;4 to 4


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1

r y.l/ 0 ∑ .n/ ( y .n ; l/ +here l 0 ;4 to 3 4

n 0 ;)

Ho+ .n/ 0 1, 1, , 1< and y .n/ 0 , ;4, ;), 1<

↑

↑

R y .;4/ or I 0;4, n 0 ;) to 31

n 0 ;) ;1 1

.n/ ( y .n ; l/

r y.;4/ 0 .;)/ (y.;) 34 / 3 .;1/( y .;1 34/ 3 ./ ( y. 34/ 3 .1/( V.134/

.;)/( V.1/ 3 .;1/ y .3)/ 3 ./y.34/ 3 .1/( V./

1 ( 1 3 1 ( 3 ( 3 1 (

0 1 3 3 3 0 1


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r y . 1/ or I 01, n 0 ;) to 31


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r y.1/ 0 .;)/ y.;) ;1 / 3 .;1/ y .;1 ;1/ 3 ./y. ;1/ 3 .1/( V.1;1/

0 .;)/ y.;4 / 3 .;1/ y .;)/ 3 ./y.;1/ 3 .1/( V./

1 ( 3 1 ( 3 ( ;4 3 1 (;) 3 3 ;) 0 )

Similarly r y.)/ 0 ;4 and r y.4/ 0

Ry.l/ 0 1, ;1, ;A, ), ), ;4, < ↑

Simple method to calculate correlation

Solution using property o correlation


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r y .l / 0 .n/ X y .;n/ ( 3 convolution of (n) and ! (-n))

; So taGe .n/ as it is: .n/ 0 1, 1, , 1<↑

; Fold y .n/: y .n/ 0 , ;4, ;), 1<

↑

y .;n/ 0 1, ;), ;4, /

↑

E=tain the con%olution o .n/ and y .;n/ =y matri method

y.;n/ .n/ 1 1 1

1 1 1 1

;) ;) ;) ;)

;4 ;4 ;4 ;4

r y.l/ 0 .1, 1 ;), ;);4, 13;43, ;)33, ;43, <

r y.l/ 0 .1, ;1, ;A, ), ), ;4, <

↑

Determine the autocorrelation o the ollo+ing signal


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.n/ 0 .1, ), 1, 1< $et 1 .n/ =e 1, ), 1, 1<

↑ ↑

)

.n/ =e 1, ), 1, 1<

Folding ) .n/ ↑

).;n/0 1, 1, ), 1<

↑

So r .l/ 0 .n/ X .;n/

↓

1.n/ 1, ), 1, 1

).;n/ 1 1 ) 1 1

1 1 ) 1 1

) ), , ), ) → 1 1, ), 1, 1

r .I/ 0 1, ) 31, 13)3), 131331, 1 3)3), )31, 1<R .l/ 0 1, 3 4, 3A, 3 5, 3 A, ,3 4, 31 <

↑

roperties o Correlation


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1( The result o autocorrelation is maimum

+hen the signal matches +ith itsel and there is no phase shiting(

)( #uto;correlation is an e%en unction

r .l/ 0 r .; l/

4( The cross;correlation is not commutati%e(

That means r y.l/ ≠ r y .l/

FIR and IIR

From the con%olution sum:


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∞

y .n/ 0∑

.G/( h .n ; G/

G 0 ; ∞

h .G/ is the impulse response o the system

FIR

I h .G/ is o inite duration, the system is called

Finite Impulse Response .FIR/ system

IIR

I h .G/ is o ininite duration, the system is called

Ininite Impulse Response .IIR/ system


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!HD

)( .t/ 0 sin.2 πt/ 3 4 sin .5) πt/ is sampled +ith Fs 0 6 times per sec.


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.1/ 7hat are the re"uencies in radians in the resulting DT signal .n/8

.)/ I .n/ is passed through an ideal interpolator,

+hat is the reconstructed signal8

To ind DT signal re"uencies sample the CT signal

ut t 0 n Ts 0 n > Fs 0 n>6 Ts sampling time period

Fs sampling re"uency 0 1> Ts

. t/ 0 sin . 2π

t / 3 4 sin .5)π

t/

? n@ 0 sin . 2 π n > 6/ 3 4 sin .5) π n> 6/

0 sin . (2 π n / 3 4 sin . 1() π n /

.* π n 3 (* .) π n 3 -. π n

0 sin .(2π

n / 3 4 sin .;(2π

n/ 0 ; ) sin .(2π

n/

0 ; ) sin .+ n/

thus + 0 2 radians

introduction to discrete time signals & system

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